An unmet demand for high resolution 3D imaging modalities providing enhanced soft tissue contrast exists in a number of biomedical disciplines. X-ray phase contrast (PC) methods can provide a solution: contrast is driven by the refractive index decrement from unity (RID) rather than the absorption term, the former being much larger than the latter for low absorbing materials and energies typically used for biomedical imaging. However, most existing PC methods suff er from drawbacks a ffecting their implementation outside specialised facilities such as synchrotrons and therefore their applicability to biomedical research. The Edge Illumination (EI) PC method has the potential to overcome or at least mitigate most of these drawbacks. Its major strengths are its simple setup, compatibility with laboratory-based x-ray sources and potential for low-dose imaging. Until now, however, EI had not been implemented in computed tomography (CT) mode. The work presented in this thesis bridges this gap. Practical and theoretical requirements for the CT extension were identifi ed, and experimental EI-CT setups were implemented both at a synchrotron and in a standard laboratory. It was shown that EI-CT allows the reconstruction of maps of a number of physical quantities, including the RID and the absorption term. Quantitative imaging via the reconstruction of the RID was demonstrated to be feasible, and to be limited only by physical eff ects not accounted for in the phase retrieval model (namely coherence) and, to some extent, polychromaticity. An optimal sampling scheme was derived for laboratory-based EI-CT. This was demonstrated to be compatible with low-dose scans, which is important for in vivo imaging. The dose such a system would deliver to a small animals (e.g. mice) was estimated, and found to be within acceptable limits. Moreover, new scientifi c areas that can benefi t from the method were identi fied (e.g. regenerative medicine), and the fi rst ever PC-CT images of acellular organ sca ffolds are presented.